ISSN 1000-9825, CODEN RUXUEW E-mail: jos@iscasaccn Journal of Software, Vol18, No4, April 2007, pp842 849 http://wwwjosorgcn DOI: 101360/jos180842 Tel/Fax: +86-10-62562563 2007 by Journal of Software All rights reserved 1,2, 1,2+, 1,2 1,2, 1 (, 100080) 2 (, 100049) Levelset Method with Multi-Speed-Function and Its Application in Segmentation of Medical Images CHEN Jian 1,2, TIAN Jie 1,2+, XUE Jian 1,2, DAI Ya-Kang 1,2 1 (Medical Image Processing Group, Key Laboratory of Complex Systems and Intelligence Science, Institute of Automation, The Chinese Academy of Sciences, Beijing 100080, China) 2 (Graduate School, The Chinese Academy of Sciences, Beijing 100049, China) + Corresponding author: Phn: +86-10-82618465, Fax: +86-10-62527995, E-mail: tian@doctorcom, http://wwwmitknet Chen J, Tian J, Xue J, Dai YK Levelset method with multi-speed-function and its application in segmentation of medical images Journal of Software, 2007,18(4):842 849 http://wwwjosorgcn/1000-9825/18/ 842htm Abstract: All of the former level set algorithms have only one level set function and only one speed function, and it is a complex procedure to minimize the energy function during the evolvement of the zero-level-set Furthermore, there are a lot of problems in this single speed function In this paper, a new multi-level-set algorithm with multiplicate speed functions is proposed according to the different properties of different objects: Different level set functions are constructed in different regions, and so are different speed functions accordingly; many zero-level-sets are evolved at the same time and act on one another in order to segment This method not only enhances the accuracy of segmentation, but also solves the bounder gap problem well, which is quite a puzzle for single level set algorithm Perfect results are achieved when this method is applied to segment the MR and CT images Key words: level set; image segmentation; speed function; energy function; MRI (magnetic resonance image) :,,, :, Supported by the National Natural Science Foundation of China under Grant Nos30600151, 30500131, 60532050 ( ); the National Grand Fundamental Research 973 Program of China under Grant No2006CB705700 ( ); the National High-Tech Research and Development Plan of China under Grant No2006AA04Z216 ( (863)); the National Science and Technology Support Program of China under Grant No2006BAH02A25 ( ); the Joint Research Fund for Overseas Chinese Young Scholars under Grant No30528027 ( ); the Beijing Natural Science Foundation of China under Grant Nos4051002, 4071003 ( ) Received 2006-04-18; Accepted 2006-05-30
: 843,,, MRI CT, : ; ; ; ;MRI(magnetic resonance image) : TP391 : A,,, [1],20,, :(1) ; (2) ;(3), [2] 1988 1995 Malladi [3],, [4 7],,, [8 11] [12 16] :,,, :(1) F,,,,,,, ;(2),,,,,, [17],, ( ), [18],,,,,,,,,, 3 1 ( ) n Γ, ω,
844 Journal of Software Vol18, No4, April 2007 ( ) 1988,Osher Sethian [2] φ(x,t), φ(x,t)=0, Γ φ(x,t) : X ω,φ(x,t)>0; X ω,φ(x,t)<0; X ω,φ(x,t)=0, Γ φ(x,t)=0, ( ),,,,,,, 1 Fig1 Multi-Level-Set method 1 : > 0, X ω1 & X Γ 1 φ 1( X, t) < 0, X ω1, F1 = 0, X Γ 1 > 0, X ω n & X Γ n φ < n( X, t) 0, X ωn, Fn = 0, X Γ n :Γ i (i=1,,n) i ;ω i (i=1,,n) Γ i,,, 2 21 [2,3] Ω, Ω ω, ω Ω,ω Γ : Γ φ(x,y), : Γ={(x,y) φ(x,y)=0} (2) Γ,Γ Γ t,φ(x,y) φ(x,y,t) t=0,γ 0 φ(x,y,t) :,d p(x,y) Γ t p(x,y) Γ t ( ), [19] φ t (x,y)=φ(x,y,t)=±d (3) (1)
: 845 Γ F [2] : φ = F φ t,f Γ,F Malladi [3],F F, (3) Hamilton-Jacobi 22, ( ), Ω, ω 1, ω 2, ω 1 ω 2 =, ω 1 ω 2 ω 1 Γ 1,ω 2 Γ 2 (3),, : φ φ 1 2 = F1 φ1, = F2 φ 2 t t F 1 F 2,,,, 23,,,,,, F 1, F 2 (φ(x,t)=0), :(1) ;(2) (1),, 0 ; (2), 0,,, Γ =Γ 1 Γ 2, Γ, Γ 1,Γ 2, 3 : (a) 1), Γ :Γ =Γ 1 Γ 2,,F Γ, 2 2), 3),, ;, (c), Fig2 Interaction between zero level sets, 2? (b) (4) (5)
846 Journal of Software Vol18, No4, April 2007 3 31, : µ ν, 3,,,, ;, 3 3,,, ;,, (a) (c) (e) µ,ν : Fig3 Segment using level set method and our method 3 µ = var( ρ ω ( ρ ) > ε ( ρ ω ), ν = num( ( ρ ω ω ) ) > ε ) µ ω ;ν ω ε :num( ) ( ) ; µ,ν, R,B R Γ Γ ;B Γ ;K Γ F : (b) (d) (f) F=(R,B,K,µ,ν) (7) µ ν,µ,ν, F, (7) F = f µ ) f ( ν ) F ( R, B, ) (8) 1( 2 K,f 1 ( ) f 2 ( ), ƒ 1 (µ)=µ+c,ƒ 2 (ν)=ν+c,,c 0 µ,ν 0, 0, 0R Γ ( 8 ) Γ, : R p = hρ q) q ω n Υ p (6) ( (9) :ω n p ;h,h ρ ρ, ρ q q R p [0,1]B [19] : B= G σ *Ω (10) :Ω ;* ; ;G σ σ K :
: 847 2 2 φ φxxφ y 2φ xφ yφxy + φ yyφx K = div 2 2 15 φ = ( φx + φ y ), R B F ( R, B, K) = e e ( F0 + F( K)) :F 0 ;F(K) K K Malladi [3], [19] 32 3, 3(c) 3(d) 3(d), [19],, ; 3(f),, 3, 4 ( 4(b) ), ; ( 4(c) 4(d) ), : 4(c) ; 4(d), 4(d), (11) 4 (a) (b) (c) (d) Fig4 Segment using level set method and our method, : 5 DWI,,,, 5(c) 5(d), 5(e) 5(f), 5(g) 5(h) 5, (a) (c) (e) (g) (b) (d) (f) (h) Fig5 Results of level set method, manual segmentation and our method 5
848 Journal of Software Vol18, No4, April 2007 6 CT ; 7 (adaptive maximum a posteriori, AMAP) ( 7(b) ) ( 7(c),AMAP ; 8 T1WI (a) (b) (c) Fig6 Results of fast matching method and our method 6 Fast matching (a) (b) (c) Fig7 Results of adaptive MAP method and our method 7 Fig8 Segmentation of white matter using our method 8 4,,,,,,, ( ),,,,,, References: [1] Hadziavdic V A comparative study of active contour models for boundary detection in brain images [PhD Thesis] University of Tromso, 2000
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